Midterm 1 Feb 21, 2014 Math 484 25 questions, 2 pts each. Section 002.Version B

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1. 2x - 1 is

(A) a linear equation for x, y,** (B) **an affine function of x,y (C) a linear form in x,

(D) none of the above.

2. x + y ≥ 0 is

**(A) a** linear constraint for x, y, (B) a linear equation for x in standard form,

(C) a linear form in x, y, (D) an equation which cannot be solved, (E) none of the above.

3. The linear program 2x+y -> max subject to x+y=1

**(A) **is unbounded, (B) is infeasible, (C) has many optimal solutions,

(D) has optimal value 3, (E) none of the above.

4. The optimal value for the linear program ax + b-> min, 0 ≤ x ≤ 1 with an unknown x and given numbers a, b is

(A) it is not a linear program , (B) the program unbounded,

**(C) **b or a + b, ** ** (D) none of the above.

5. The mathematical program x/y ->max subject to |x| ≤ 1, y ≥ 1 for two unknowns x, y

(A) is a linear program, (B) is infeasible,** (C) h**as an optimal solution,

(D) **i**s unbounded, (E) is not a mathematical program.

6. The mathematical program x -> max, x ≤ 1

(A) is not a linear program, (B) is infeasible, (**C) h**as an optimal solution,

(D) is unbounded,** **(E) none of the above.

7. The answer to the mathematical program 4x+3y -> min , x^{2}+ y^{2} ≤ 25 is

(A) max= 25 at x = 4, y = 3, (B) max = 30 at x = 4, y = 3, (C) max = 50 at x = 5, y =5,

**(D)** min =-25 at x = -4, y = -3, ( E) none of the above.

8. The minimal total time for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

3 1 0 0

is ** **(A**)** 4, (B) 6, (C) 7, (D) 8, **(E) **none of the above.

9. For each number t, the equation cos(t)x= sin(t) for unknown x

(A) is not linear, (B) has a solution, **(C) **is a linear equation, (D) has many solutions, (E) none of the above.

10. 0 ≤ 0 provided that x = y. **A) **True, B) False

11. x > 0 and y > 0 if xy < 0. A) True, ** B) **False

12. x > 0 only if x ≥ 2. ** **A) True,** B) **False

13. The linear program given by a standard tableau with the matrix

-1 |
-2 |
-1 |
-3 |
0 |

-4 |
0 |
-2 |
0 |
1 |

0 |
1 |
0 |
2 |
-3 |

(A) is infeasible, (B) is unbounded, **(C) **has an optimal solution,

(D) has only 2 linear constraints, (E) none of the above.

14. The linear program given by the standard tableau

x1 x2 x3 x4 1

1 2 0 -3 0 =x5

4 0 -2 0 1 =x6

0 -1 0 2 -3 -> min

(A) is infeasible, **(B)** is unbounded., (C) has an optimal solution,

(D) requires at least one pivot step to solve by simplex method.

15. The linear program given by a standard tableau with the matrix

1 |
2 |
0 |
-3 |
0 |

-4 |
0 |
-2 |
0 |
1 |

0 |
-1 |
0 |
2 |
-3 |

(A) is infeasible,** (B)** is unbounded, (C) has an optimal solution,

(D) has -3 as the optimal value, (E) has only one feasible solution.

16. The standard tableau

x1 x2 x3 1

-1 0 0 0 -> min

(A)** **is optimal, **(B)** has a bad column, (C) has a bad row, **(D) i**s not standard, (E) none of the above.

17. The standard tableau

1

-1 = x1

0 = x2

-1 -> min

(A) is optimal. (B) has a bad column, **(C) h**as a bad row, (D) is not standard, (E) is too small.

18. Pivoting the tableau

x -y

2* 3 = z

produces the tableau

**A) **z -y B) x y

1/2 -3/2 = x 2 -3/2 = z

C) x y D) z y E) x y

1/2 3 = z 1/2 3/2 =x 1/2 -3/2 = z

19. The linear program x+y -> max, x≤ -1, y < -2

A) is infeasible.** B**) is bounded. C)** h**as an optimal solution.

D) has infinitely many optimal solutions. E) has 1 as the optimal value.

20.The equation x^{3}+ y^{3} ^{ }= 0 for x, y

A) has no solutions, B) has exactly one solution, **C) **has infinitely many solutions.

21.The system x+ 2y = 3, 4x + 8y = 1 for x,y

**A)** has no solutions. B) has exactly one solution. C) has infinitely many solutions.

22. The set of constraints x + y > 2, 2x + 3y > 4 implies that

A) 4x + 5y ≥ 9, B) x ≥ 0, C) y ≥ 1, D) x ≤ 0, **E) no**ne of the above.

23. For any number x, 0 = 1 implies that x ≥ 2 and x ≤ 1. ** A) **True B) False

24. For any number x, if x^{2 }= 0 then x ≥ 0 or x ≤ 0. ** ** **A) **True B)False

25. For any column x of numbers, if 0 = 0 then x ≥ 0 or -x ≥ 0. A) True **B)** False